2.13A Logarithmic and Exponential Equations and Inequalties Part A

Introduction to Exponential and Logarithmic Equations

  • Overview: Discuss exponential and logarithmic equations/inequalities in a two-part series.

    • Focus on equations in part one.

    • Set up anticipation for part two.

Logarithmic Properties

Basic Logarithmic Principles

  • Equality of Logarithms: If ( \log_a(b) = \log_a(c) ), then ( b = c ) (if all logs share the same base).

  • Example of Addition in Logs:

    • If ( \log_a(x) + \log_a(y) = \log_a(z) ), it can be rewritten using the product property: ( \log_a(xy) = \log_a(z) ).

Example Problem 1: Logarithmic Addition

  • Given: ( \log_3(x) + \log_3(x + 7) = \log_3(60) )

  • Reformulate using Product Property: ( \log_3(x(x + 7)) = \log_3(60) ) implies ( x(x + 7) = 60 ).

  • Solve: Form the quadratic equation ( x^2 + 7x - 60 = 0 ).

  • Factoring: Find factors of -60 that add to 7: ( (x + 12)(x - 5) = 0 ).

    • Solutions: ( x = -12 ) (extraneous, invalid for logs), ( x = 5 ).

  • Check Solution: Substitute ( x = 5 ) back into the original equation, confirming validity.

Example Problem 2: Logarithms and Constants

  • Given: ( 3 = \log_2(x + 7) + \log_2(5) )

  • Combine Logs: ( 3 = \log_2(5(x + 7)) )

  • Convert to Exponential Form: ( 2^3 = 5(x + 7) ) leads to ( 8 = 5(x + 7) ).

  • Isolate x: ( x = -\frac{27}{5} ); check to confirm logarithms are valid (logarithm of positive quantity).

Example Problem 3: Solve Exponential Equations with Logs

  • Given: ( 3 \times 2^{(x + 2)} = 12 \)

  • Divide to isolate exponent: ( 2^{(x + 2)} = 4 ).

  • Apply Logarithmic Property: Use ( \log_2(2^{(x + 2)}) = \log_2(4) ) where cancelling gives ( x + 2 = 2 ).

  • Final Solution: ( x = 0 ), ensuring both sides equal when plugged back.

Example Problem 4: Graphing Technology for Solutions

  • Using Graphing Calculator: Input equations into a graphing calculator as ( y_1 ) and ( y_2 ). Find intersection point as solution to the system of equations.

  • Example Result: Solved intersection yields ( x = -5.9 ).

Example Problems for Practice

  1. Logarithm to Exponent Equation: ( e^4 = \sqrt{x - 12} )

  • Transform and solve: ( e^8 = x - 12 \implies x = e^8 + 12 \approx 2992.96 ) (depending on calculator allowance).

  1. Base Conversion and Logarithmic Comparisons: Solve equations formulating comparisons between logarithmic bases.

  2. Combining logs: Logarithmic properties to combine terms, then solve for ( x ).

Conclusion

  • Anticipation for part two is built upon mastering concepts from part one.

  • Encouragement to engage with practice problems and reinforce understanding before moving on.